Beat frequencies/tension force of violin strings while tuning-

[frozenguy]

Beat frequencies/tension force of violin strings while tuning--please help

Homework Statement

To tune your violin, you first tune the A string to the correct pitch of 440 Hz, and then you
bow both it and the adjoining string simultaneously, all the while listening for beats. While
bowing the A and E strings, you hear a beat frequency of 3.00 Hz and note that the beat
frequency increases as the tension on the E string is increased. (The E string is to be tuned to 660 Hz)
(a) Why are the beats produced?

Homework Equations

The Attempt at a Solution

I wrote that the beats were caused due to constructive/destructive interference, but the answer is that "The fundamentals of A and E differ too much to cause a beat frequency of 3hz, therefore it must be caused by the harmonics of A and E."

It then says " So the E string must be slightly greater then 660hz because the beat frequency increases as the tension force increases."

I understand that the beat frequency increases as the tension force increases, maybe not why, but ok, I hear it, but I don't get how he figured out the E string must be slightly greater then 660hz..

How can you tell the beat freq increased at all? I don't really get how he came to that conclusion.

What if it was lower then 660hz? What would the beat freq look like?

Also, what really is a fundamental or harmonic? I know what it means in terms of calculating stuff, but what is it? Is is different sections of the wave? Is it cycles of the wave? For every wave is there a fundamental wave, a harmonic wave (1,2,3,4 and so on)? I just don't get what they are in essence I guess.

 

[ideasrule]

I understand that the beat frequency increases as the tension force increases, maybe not why, but ok, I hear it, but I don't get how he figured out the E string must be slightly greater then 660hz..

As tension increases, the speed of sound increases, so frequency has to increase as well. Since the beat frequency increased, that means increasing the frequency gets you farther from 660 Hz.

How can you tell the beat freq increased at all? I don't really get how he came to that conclusion.

That's what the question says.

Also, what really is a fundamental or harmonic? I know what it means in terms of calculating stuff, but what is it? Is is different sections of the wave? Is it cycles of the wave? For every wave is there a fundamental wave, a harmonic wave (1,2,3,4 and so on)? I just don't get what they are in essence I guess.

It's how the string is vibrating. If the string's vibrating in the same direction, that's the harmonic. If the string is vibrating as a complete wave and takes the shape of a sine wave, that's the second harmonic. If the string has 1.5 complete waves, that's the third.

 

[frozenguy]

As tension increases, the speed of sound increases, so frequency has to increase as well. Since the beat frequency increased, that means increasing the frequency gets you farther from 660 Hz.

That's what the question says.

It's how the string is vibrating. If the string's vibrating in the same direction, that's the harmonic. If the string is vibrating as a complete wave and takes the shape of a sine wave, that's the second harmonic. If the string has 1.5 complete waves, that's the third.

Thanks so much! I'm still a little confused..

The beat frequency can't be negative right? So how do we know its not less then 660? what did the beat frequency increase from to get to 3? If its any positive number its about 660? I just don't get that..

So can a string vibrate with all the harmonics at once? When you pluck a string, does it start vibrating at some harmonic and start changing numbers as it dies down?

 

[ideasrule]

Yes, the beat frequency can't be negative; it's just the difference between two frequencies.

The question says the beat frequency was 3, THEN it increased after tension was increased. So there are two possibilities. You know the A string is at 660 Hz (well, one of its harmonics is), so the E string can either be at 657 or 663 Hz. If you increase the E string's frequency by, say, 1 Hz, it will be at either 658 or 664 Hz. 658 Hz is closer to 660 Hz; 664 is farther.

 

[Pythagorean]

So can a string vibrate with all the harmonics at once? When you pluck a string, does it start vibrating at some harmonic and start changing numbers as it dies down?

The harmonics are pretty much constant as the string rings. The higher the harmonic, the softer it is though, so naturally you hear the fundamental the loudest.

Here's an example of the fundamental and it's harmonics. All these waveforms (and even higher order ones) exist on the plucked string at once. The first one is the fundamental:

This picture is from the wiki on the harmonic series (music):
http://en.wikipedia.org/wiki/Harmonic_series_(music)

There's a mathematics harmonic series wiki too

 

[Stonebridge]

The fundamental of the A string is 440Hz
The 2nd harmonic (1st overtone) is 880Hz
The 3rd harmonic (2nd overtone) is 1320Hz
These are whole multiples of 440

The fundamental of the E string is 660Hz
The 2nd harmonic (1st overtone) is 1320Hz
These are whole multiples of 660

The beats occur between the harmonics at 1320Hz.

The beat frequency of 3Hz means, in this case, that the E string 2nd harmonic is at 1323Hz - slightly sharp. I know this because increasing the tension in that string makes it slightly sharper and the beat frequency increases.

Musically, you hear the 2nd harmonic as the note an octave above the fundamental.
The 3rd harmonic is a 5th above that.
So in this case, the beats are between two E notes.
The one is the octave above the E string. (2nd harmonic on that string)
The other is the E which is the 3rd harmonic on the A string.

 

[frozenguy]

Ok.. I'm understanding it much more now guys.. Thanks a lot :)